# If $a = b$ then $b=a$… Flipping lhs and rhs of algebraic equations

So I'm reading through Introductory Algebra for College Students and in one of the side notes the author has a question am I allowed to 'flip' the sides of an equation. Assuming a and b are real. If $a = b$ then $b=a$.

I'm having trouble with this statement... Can you flip the left hand side and right hand side as long as you keep the numbers in the same order?

Can anyone find some examples that might help me wrap my head around this? Visual explanations are appreciated

• The equals sign in mathematics means that the two expressions are essentially different names for the same thing. So you can switch left and right sides whenever you like. See math.stackexchange.com/questions/2361063/… – Ethan Bolker Jun 2 '18 at 16:19
• This is the symmetry property of equivalence relations – qwr Jun 2 '18 at 16:33

Flipping sides of the equation is perfectly fine as long the statements on both sides follow Communicative, Associative, and Distributive rules for their respected operators.

Example:

$4x + 2 = 8$

is also

$8 = 4x + 2$

which is also

$8 = 2 + 4x$

because addition holds the commutative property.

and

$8 = 2 + x4$

Because multiplication also holds the commutative property.

Example 2:

$3(4+1) = 15$

is also

$15 = 3(4+1)$

which is also

$15 = (12 +3)$

because Multiplication holds the distributive property.

Keeping your numbers in order will save some headache and copy errors, but my point is that you should know that equality of both sides of a statement lies within the rules each operator follows.

Well, lets say you are evaluating how much money you have vs how much money your friend has. You have 4 - \$5 dollar bills. Your friend has 2 -$10 bills. After thinking about it, you conclude you have the same amount of money because:

$$4 \cdot \5 = 2 \cdot \10$$

With the above 'flipping' you describe, we can immediately state:

$$2 \cdot \10 = 4 \cdot \5$$

The idea of one thing equalling another simply means they are saying the same thing in two ways. Thus it's perfectly valid to flip the terms over.